Extension:MathJax/ru

http://traditio-ru.org/wiki/MathJax_для_MediaWiki

http://traditio-ru.org/wiki/%D0%A2%D1%80%D0%B0%D0%B4%D0%B8%D1%86%D0%B8%D1%8F:%D0%9F%D1%80%D0%B8%D0%BC%D0%B5%D1%80%D1%8B_%D0%BE%D1%84%D0%BE%D1%80%D0%BC%D0%BB%D0%B5%D0%BD%D0%B8%D1%8F_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB

http://traditio-ru.org/wiki/Справка:Формулы

http://people.cs.kuleuven.be/~dirk.nuyens/Extension_MathJax/

http://www.mediawiki.org/wiki/Extension:MathJax/ru

$

 \newcommand{\Re}{\mathrm{Re}\,}
 \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}

$

We consider, for various values of $s$, the $n$-dimensional integral \begin{align}

 \label{def:Wns}
 W_n (s)
 &:= 
 \int_{[0, 1]^n} 
   \left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x}

\end{align} which occurs in the theory of uniform random walk integrals in the plane, where at each step a unit-step is taken in a random direction. As such, the integral \eqref{def:Wns} expresses the $s$-th moment of the distance to the origin after $n$ steps.

By experimentation and some sketchy arguments we quickly conjectured and strongly believed that, for $k$ a nonnegative integer \begin{align}

 \label{eq:W3k}
 W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.

\end{align} Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. The reason for \eqref{eq:W3k} was long a mystery, but it will be explained at the end of the paper.